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Properties

Label 360.6.f.a
Level $360$
Weight $6$
Character orbit 360.f
Analytic conductor $57.738$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,6,Mod(289,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.289");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.7381751327\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.25787221056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 61x^{4} + 852x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3} + \beta_{2} + \cdots - 9) q^{5} + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots + 7 \beta_1) q^{7} + (9 \beta_{5} - 4 \beta_{4} + \cdots + 111) q^{11} + (3 \beta_{4} - 3 \beta_{3} + \cdots - 7 \beta_1) q^{13}+ \cdots + ( - 204 \beta_{4} + 204 \beta_{3} + \cdots + 322 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 50 q^{5} + 664 q^{11} + 288 q^{19} + 3750 q^{25} + 1892 q^{29} - 10248 q^{31} + 18880 q^{35} + 1324 q^{41} + 28410 q^{49} - 47160 q^{55} + 10296 q^{59} - 52116 q^{61} + 13480 q^{65} + 28288 q^{71}+ \cdots + 349120 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 61x^{4} + 852x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 85\nu^{3} + 1596\nu ) / 324 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{5} - 611\nu^{3} - 7080\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 21\nu^{4} - 263\nu^{3} + 1137\nu^{2} - 1662\nu + 8892 ) / 324 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{5} + 21\nu^{4} + 263\nu^{3} + 1137\nu^{2} + 1662\nu + 8892 ) / 324 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 37\nu^{2} + 78 ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -3\beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{5} + 9\beta_{4} + 9\beta_{3} - 403 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 97\beta_{4} - 97\beta_{3} + 39\beta_{2} + 317\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 379\beta_{5} - 333\beta_{4} - 333\beta_{3} + 13351 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3457\beta_{4} + 3457\beta_{3} - 1719\beta_{2} - 15677\beta_1 ) / 20 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.843753i
− 0.843753i
6.33429i
− 6.33429i
− 4.49053i
4.49053i
0 0 0 −42.6733 − 36.1108i 0 168.485i 0 0 0
289.2 0 0 0 −42.6733 + 36.1108i 0 − 168.485i 0 0 0
289.3 0 0 0 −33.8706 − 44.4723i 0 85.8595i 0 0 0
289.4 0 0 0 −33.8706 + 44.4723i 0 − 85.8595i 0 0 0
289.5 0 0 0 51.5439 − 21.6386i 0 − 21.3742i 0 0 0
289.6 0 0 0 51.5439 + 21.6386i 0 21.3742i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.6.f.a 6
3.b odd 2 1 120.6.f.a ✓ 6
4.b odd 2 1 720.6.f.l 6
5.b even 2 1 inner 360.6.f.a 6
12.b even 2 1 240.6.f.e 6
15.d odd 2 1 120.6.f.a ✓ 6
15.e even 4 1 600.6.a.r 3
15.e even 4 1 600.6.a.s 3
20.d odd 2 1 720.6.f.l 6
60.h even 2 1 240.6.f.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.f.a ✓ 6 3.b odd 2 1
120.6.f.a ✓ 6 15.d odd 2 1
240.6.f.e 6 12.b even 2 1
240.6.f.e 6 60.h even 2 1
360.6.f.a 6 1.a even 1 1 trivial
360.6.f.a 6 5.b even 2 1 inner
600.6.a.r 3 15.e even 4 1
600.6.a.s 3 15.e even 4 1
720.6.f.l 6 4.b odd 2 1
720.6.f.l 6 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 36216T_{7}^{4} + 225603600T_{7}^{2} + 95604640000 \) acting on \(S_{6}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 95604640000 \) Copy content Toggle raw display
$11$ \( (T^{3} - 332 T^{2} + \cdots - 751600)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 743071584256 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{3} - 144 T^{2} + \cdots + 2584131584)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{3} - 946 T^{2} + \cdots - 37303134464)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 5124 T^{2} + \cdots - 71486323200)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{3} - 662 T^{2} + \cdots - 289547160040)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots + 6990217203600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 30927698302664)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 37186025753600)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 4303737800704)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 11306161037592)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
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